The most difficult part of teaching trigonometry to kids (more specifically, pre-university and below) is proving trigonometric identities. Typically, the stronger students would relish the challenge, while the weaker would despair. Some of them would even question what is the point of all these. Another potential problem is that students are told to accept certain identities and use them. One chief culprit would be the addition formula:

[tex] \sin ( a + b) = \sin(a) \cos(b) + sin(b) \cos(a)[/tex].

In all my years of formal mathematical education, I do not recall ever seeing that identity being proved. I sort of figured it out for myself, while learning complex analysis, which to me is the best proof. One version is given here. Ah, the beauty of complex analysis!

The shortest path between two truths in the real domain passes through the complex domain.

-Jacques Hadamard (1865-1963)

More info on the above quote given here.

Of course, a purely geometric proof exists and involves drawing one triangle on top of another. A more interesting second proof uses Ptolemy’s theorem.

I’m a visual person, so I like the proof based on rotation matrices– probably because that is the easiest for me to remember. It’s clear what the images of the two axes are under rotation, and then by linearity, you get the formula. And an extra one, too.

For another proof for [tex]\sin(a+b) = \sin a \cos b + \cos a \sin b[/tex] http://www.rinconmatematico.com/senosuma/senosuma.pdf